Historical notes |
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Birkhoff and the Alexandrov specialization order |

A single axiom for closure operators (soon!) |

Lattice theory is at the basis of several scientific specialities:

- abstract interpretation of programming (semantics), in particular of distributed and parallel programming;
- modelling of imprecision (roughness) and uncertainty (fuzzyness);
- formal concept analysis;
- morphological image processing.

It finds uses in many practical activities, from discrete event systems to the struggle against terrorism.

However it has always had a bad reputation, as some kind of useless abstraction, and most scientists, especially mathematicians, don't understand it at all.

*On p. 1440:*

Never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory. Dedekind, Jónsson, Kurosh, Malcev, Ore, von Neumann, Tarski, and most prominently Garrett Birkhoff have contributed a new vision of mathematics, a vision that has been cursed by a conjunction of misunderstandings, resentment, and raw prejudice.

The hostility towards lattice theory began when Dedekind published the two fundamental papers that brought the theory to life well over one hundred years ago. Kronecker in one of his letters accused Dedekind of ``losing his mind in abstractions,'' or something to that effect.

I took a course in lattice theory from Oystein Ore while a graduate student at Yale in the fall of 1954. The lectures were scheduled at 8 a.m., and only one other student attended besides me-María Wonenburger. It is the only course I have ever attended that met at 8 o'clock in the morning. The first lecture was somewhat of a letdown, beginning with the words: ``I think lattice theory is played out'' (Ore's words have remained imprinted in my mind).

For some years I did not come back to lattice theory. In 1963, when I taught my first course in combinatorics, I was amazed to find that lattice theory fit combinatorics like a shoe.

G. C. Rota, while writing the introduction to M. Stern's
book *Semimodular lattices* [Teubner, Stuttgart, 1991;
MR1164868 (94e:06005)], mentions the following two anecdotes.

- J. von Neumann delivered an invited lecture at a weekly mathematics colloquium at Harvard sometime in the 1930s. After his lecture on `Continuous Geometries', G. H. Hardy, who was visiting Cambridge, was overheard whispering to G. D. Birkhoff (Garrett's father): ``He is quite clearly a brilliant man, but why does he waste his time on this stuff?''
- Rota himself was stopped in a hall at M. I. T. by a much senior mathematician, who, staring him in the eyes, demanded in no uncertain terms: ``Admit! All lattice theory is trivial.''

[Quoted by N. K. Thakare in *Mathematical Reviews*, MR2351371
(2008j:06009)]

*On p. 450:*

Other formal concepts may be developed ``before their time'' to find use only much later. For example, the notions of lattice theory were found about 1900 by Dedekind and others, but did not at that time find any noticeable resonance. The same notions, when rediscovered by Garrett Birkhoff and Oystein Ore in the early 1930's, were immediately put to use in projective geometries, continuous geometries, and in the analysis of subobjects of algebraic systems. It would seem that by 1930 there were at hand more uses for such abstract notions. Subsequently, after a lively decade of research, lattice theory became of less central interest to algebraic developments-it may be because the principal uses were already worked out and the remaining problems were artificial.

In 2006 F. Wehrung gave a counterexample to a half-century old
conjecture by Dilworth about congruence lattices. This problem was
considered by leading experts such as G. Grätzer as central in
lattice theory. Despite very positive reviews by referees, his paper
was rejected by the *Journal of the American Mathematical
Society* on the ground that it lacked ``interaction with other
areas of mathematics''.

See the Letters
to the Editors section in the *Notices of the American Mathematical
Society* and the opinion by Doron Zeilberger,
Because
You Snubbed Others You Were Snubbed, and Those Who Snubbed You Shall
Be Snubbed.